$S_1 = \frac{n}{2}[2a+(n-1)d]$ ; $S_2 =\frac{2n}{2}[2a+(2n-1)d]$ ; $S_3 =\frac{3n}{2}[2a+(3n-1)d]$$\...

Answer : $75^{\large\circ},85^{\large\circ},95^{\large\circ},105^{\large\circ}$

Cost of the equipment = Rs 6,00,000$a = 15\% $ d =$ -1.5\%$n =10 $\begin{align*}Sn& = \frac{1...

$1+2+3+....(x-1)=(x+1)+(x+2)+...49$$\implies 1+2+3+...(x-1) = (1+2+3+....+x+x+1+...49)-(1+2+3+....+x...

Let the number be (a-d), a, (a+d)$(a+d)\times100 + a\times10+(a-d) = 111a + 99d$on reversing$\begin{...

Sum = 3250$\frac{n}{2}[40+(n-1) \times15 ]=3250 $on simplifying we get,$\begin{align*} 15n^2 +25n-...

Sum = 280$\begin{align*}\frac{4}{2}[2n+(4-1)\times 20]& =280 \\ 2(2a-60)& = 280 \\ a& =...

Answer : $(n+1) : n$Let $a_p$ be the $p^{th}$ term of the given A.P$\therefore a_p=a+(p-1)d$$S_1$=Su...

$a_10=52$ ; $a_16 = 82$$\implies a+9d = 52$ and $a+15d = 82$on solving both the equation we get a=...

$a_7 = -1$; $a_{16} = 17$$\implies a+6d = -1$ and $a+15d =17$$\begin{align*} 9d &= 18 \\ d &a...

$ a_p = a + (p - 1)d = a$$ a_q = a + (q - 1)d = b$$ a_r = a + (r - 1)d = c$$ (a - b) = (p - q)d, \;\...

Answer : $\large\frac{7}{5}$

Answer : $\large\frac{3ab}{2(b-a)}$

$ S_n = 2n^2 + 5n$$ S_1 = a \; \; \; = 2 (1)^2 + 5 (1) = 7$$ S_2 = a + a + d \implies 2a + d$$ 2 (2)...

$ S_n = \frac {n}{2} [2a + (n - 1)d]$$d=S_n-kS_{n-1}+S_{n-2}$$ =\frac {n}{2} [2a + (n - 1) a] - k [\...

Answer : 0

Let the numbers be a-d, a, a+d$ a - d + a + a - d = 27$$ \implies 3a = 27 \implies a = 9$$ Also: \;...

$ a_n = 4n + 1$$ a_1 = 4 (1) + 1 = 5$$ a_2 = 4 (2) + 1 = 9$$ d = a_1 - a_2 = 9 - 5 = 4$$ a = 5, \; \...

$S_n=3n^2-n$$ S_1 = a \; \; \; = 3 - 1 = 2$$ S_2 = a + a + d = 2a + d = 3 (2)^2 - 2 = 10$$ \implies ...

$S_n=(3n^2+2n)$$ S_1 = a \; \; \; \; = 3 + 2 = 5$$ S_2 = a + a + d \; \; = 2a +d \; \; = 3 \times 2...

$ a + (p - 1) d = a$$ a + (q - 1) d = b$$ a + (r - 1) d = c$$ a(q -r) + b(r - p) + c(p -q)$$ =[(a + ...

$ a_1 = 94 \;\; and \; \; d = - 4$ $\begin{align*} a_15 &= a + 14 d \\& = 94 - 14 \times 4 ...

$a_1 = 2x, \; \; a_2 = (x+10)$ and $a_3 = (3x+2)$$ a_2 - a_1 = x + 10 - 2x \implies 10 - x$$ a_3 - ...

$a_1 = 24, \; \; and \; \; d = -3$$a + (n - 1)d = 0$$ 24 + (n -1) -3 = 0$$ 24 - 3n + 3 = 0$$ 27 = 3n...

$ a_p = q \; \; and \; \; a_q = p$$(i,e) \; \; a + (p -1) d = q \; \ and \; \; a + (q - 1) d = p$On...

$ a = 5a - x$$ d = 6a - (5a -x) \implies a+ x $$ \begin{align*} a_{11} &= a + 10d \\& = 5a -...

$ m.a_n = n.a_n$$(i.e) \; m (a + (n-1)d) = n (a + (n-1)d)$$ \implies m (a + (n-1)d) - n (a + (n-1)d)...

Answer : $\large\frac{13}{\sqrt 2}$

$ A_1 = 4 + 3d$$ A_2 = 9 + 3d$$ A_2 - A_1 = 9 + 3d - 4 + 3d$$ Ans = 5$

$ a_3 = 18 \; \; and \; \; a_7 = 30$$\implies a + 2d = 18 \; \; and\; \; a + 6d = 30$On solving the...

Answer : twice

$ a_3 = p \; \; and \; \; a_4 = q$$ (i.e) a + 2d = p \; \; and a + 3d = q$$ \implies d = q - p \; \...

Answer : twice

$ 7a_7 = 11a_{11}$$ (i.e) 7 (a + (7-1)d) = 11 (a + (11-1) d)$$\implies 7a + 42d = 11a + 110d$$ \impl...

$S_n = xm^2 + ym$$ S_1 = a = x(1) + y(1) = x + y $$ S_ 2 = a + a + d = 2a + d$$ = x(2)^2 + y(2) = 4x...

$ a_1 = \frac {1}{11}, \; \; a_n = \frac{17}{11} \; \; and \; \; d = \frac {1}{11}$$ a_n = a + (n - ...

$a_1 = -1, \; \; a_n = 95 \; \; and \; \; d = 4$$ a_n = a + (n - 1)d$$ 95 = -1 + (n - 1) \times 4$$ ...

$a_1 = 12, \; \; a_n = 99 \; \; and \; \; d = 3$$ a_n = a + (n - 1)d$$ 99 = 12 + (n - 1) \times 3$$ ...

$ a_1 = 10, \; \; a_n = 95 \; \; and \; d = 5$$ a_n = a + (n - 1) d$$ 95 = 10 + (n - 1) \times 5$$ ...

$ a_n = 5n - 7$$ a_1 = 5(1) - 7 = -2$$ a_2 = 5(2) - 7 = 3$$ d = a_1 - a_2 \; \; \implies 3 - (-2) = ...

Answer : $\large\frac{n-1}{4}$

$ a = -3 \; \; and \; \; d = 5 - (-3) = 8$$ a_{25} = -3 + (25 - 1) \times 8 = 192 - 3$$ = 189$

$a_1=2x, a_2 =x+10, a_3= $3x+2$$a_2-a_1 = a_3 - a_2$$\begin{align*}a_2 - a_1 & = a_3-a_2 \\ \imp...

$a_1=\sqrt 2$ , $a_2 = \sqrt8=2\sqrt2 $, $ a_3 = \sqrt15= 3\sqrt2$d= $a_2 - a_1$d = $2\sqrt2 - \...

$a_1 =(x+y), a_2 = (x+1)+y, a_3 = (x+1)+(y+1),$$\begin{align*}a_2 - a_1 & = d \\ \implies d &...

$a_1 = 3 - \sqrt7,a_2 =2-\sqrt 7,a_3=1-\sqrt 7$$\begin{align*} d & ={a_2 - a_1} \\ & = 2 - \...

Answer : $\large\frac{\sqrt y}{1-y}$

x - y = 2cm $\implies x = y+2$hypotenuse = $10$ cm$(y+2)^2 + y^2 = 10^2$$\implies y^2 + 4y + 4 + y^2...

Answer : $(3-\sqrt 5)$ m

Given: Area of the triangle = $18cm^2$Let the base & height be x and y respectively$ x + y = 12 ...